Jie Liu on projective space

Exciting news!  Jie Liu has proven a conjecture of mine -- thereby resolving an old conjecture of Sommese.  I'll briefly explain his result in this post.

The conjecture was:

Conjecture (L---, now proven by Jie Liu).  Let \(X\) be a smooth projective variety over \(\mathbb{C}\), and \(\mathscr{E}\) an ample vector bundle on \(X\).  Suppose that $$\text{Hom}(\mathscr{E}, T_X)\not=0.$$ Then \(X\) is isomorphic to \(\mathbb{P}^n\) for some \(n\).  

This conjecture was known (due to Andreatta-Wisniewski) in the case that there was a map \(\mathscr{E}\to T_X\) of constant rank; Liu in fact proves something slightly stronger:

Theorem (Liu).   Let \(X\) be a smooth projective variety over \(\mathbb{C}\), and suppose \(T_X\) contains an ample subsheaf \(\mathscr{F}\).  Then \(X\simeq \mathbb{P}^n\) for some \(n\).

The proof is quite nice.  Old results of Araujo, Druel, and Kovacs immediately show that there exists an open subscheme \(X_0\subset X\) and a map \(X_0\to T\) whose fibers are isomorphic to \(\mathbb{P}^r\) for some \(r\).  We wish to show that \(T\) is a point.  (Up to this point, this more or less follows Andreatta-Wisniewski).  This is not particularly hard if the \(\mathbb{P}^r\)-bundle \(X_0\to T\) is defined away from codimension two; indeed, in this case, one may find a proper curve in \(T\) and obtain a contradiction using standard techniques originally due to Peternell-Campana in this case -- this is essentially how Andreatta-Wisniewski proceed.  The key fact is that if \(f: Y\to C\) is a \(\mathbb{P}^r\)-bundle over a proper curve \(C\), the relative tangent sheaf \(T_f\) contains no ample subsheaves.

Unfortunately if \(\mathscr{F}\) is not locally free, I don't see how to extend the fibration to codimension two, so this approach seems to be sunk.

Liu cleverly gets around this issue by working on an open subset of \(X\) rather than a closed subset.  Unfortunately the paper is quite densely written and I do not as yet understand all the details -- if someone would like to explain what is happening "morally," I would love to hear it.

In any case, the results of the paper in which I made this conjecture allow one to deduce from this result an old conjecture of Sommese, which classifies smooth varieties containing a projective bundle as an ample divisor.  This conjecture has attracted a lot of interest over the last few decades, so it's very gratifying to see it put to rest.